Positive and nodal solutions for parametric nonlinear Robin problems with indefinite potential

Genni  Fragnelli; Dimitri  Mugnai; Nikolaos S. Papageorgiou

doi:10.3934/dcds.2016068

Article Contents

2016, Volume 36, Issue 11: 6133-6166. Doi: 10.3934/dcds.2016068

This issue Previous Article Quasi-stability property and attractors for a semilinear Timoshenko system Next Article An infinite-dimensional weak KAM theory via random variables

Positive and nodal solutions for parametric nonlinear Robin problems with indefinite potential

1.
Department of Mathematics, University of Bari Aldo Moro, Via E.Orabona 4, 70125 Bari
2.
Department of Mathematics and Computer Sciences, University of Perugia, Via Vanvitelli 1, 06123 Perugia
3.
Department of Mathematics, National Technical University, Zografou Campus, Athens 15780

Received: November 2015

Revised: April 2016

Published: May 2017

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Abstract

We consider a parametric nonlinear Robin problem driven by the $p -$Laplacian plus an indefinite potential and a Carathéodory reaction which is $(p-1) -$ superlinear without satisfying the Ambrosetti - Rabinowitz condition. We prove a bifurcation-type result describing the dependence of the set of positive solutions on the parameter. We also prove the existence of nodal solutions. Our proofs use tools from critical point theory, Morse theory and suitable truncation techniques.

Keywords:

Indefinite potential,
superlinear reaction,
nonlinear regularity and maximum principle,
Picone identity,
bifurcation-type theorem for positive solutions,
nodal solutions,
critical groups.

Mathematics Subject Classification: Primary: 35J20, 35J60; Secondary: 58E05.

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References

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